Question

A wire of uniform cross-section \(A\), length \(l\) and resistance \(R\) is bent into a complete circle. The equivalent resistance between any two diametrically opposite points will be:

• A. \(\dfrac{R}{4}\)
• B. \(\dfrac{R}{8}\)
• C. \(4R\)
• D. \(\dfrac{R}{2}\)

Hint:

For a circular wire of total resistance \(R\): \[ \text{Diametrically opposite points} \Rightarrow \frac{R}{2} \parallel \frac{R}{2} \] Therefore, \[ R_{\text{eq}} = \frac{R}{4} \] This is a very common CUET/JEE/NEET formula-based question.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The electrical resistance ($R$) of a uniform conductor is directly proportional to its continuous path length ($l$), as defined by the relationship $R = \rho \frac{l}{A}$. When a straight wire is reshaped or bent into a circle, its total resistance splits across separate arc segments based on how you hook up external terminal connections to the loop.
Step 2: Key Formula or Approach:
Connecting terminals to diametrically opposite points divides the circle into two identical semicircular paths. - Because each semicircular path contains exactly half the wire's total arc length ($\frac{l}{2}$), each half-loop path carries exactly half the original resistance: $$ R_1 = R_2 = \frac{R}{2} $$ These two semicircular branches are configured in a parallel combination relative to the two connection points. The equivalent parallel resistance formula for two resistors is: $$ \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} \implies R_{\text{eq}} = \frac{R_1 \times R_2}{R_1 + R_2} $$
Step 3: Detailed Explanation:
Let's analyze the parallel resistance branch circuit: 1. The top semicircular branch has a resistance value of: $R_1 = \frac{R}{2}$ 2. The bottom semicircular branch has an identical resistance value of: $R_2 = \frac{R}{2}$ Since both parallel resistors have the exact same resistance value, their total equivalent resistance ($R_{\text{eq}}$) is simply equal to half the value of one individual branch: $$ R_{\text{eq}} = \frac{\text{Branch Resistance}}{2} $$ $$ R_{\text{eq}} = \frac{\left(\frac{R}{2}\right)}{2} = \frac{R}{4} $$ This algebraic reduction matches option (A).
Step 4: Final Answer:
The equivalent resistance between the two diametrically opposite points is R/4.